Estimating total mass inside the Sun's orbit

Brad 05/13/2018. 1 answers, 32 views
homework-and-exercises dark-matter galaxies milky-way galaxy-rotation-curve

I'm trying to estimate the local density of dark matter at a given radius, $r=R_0=8\text{kpc}$. I also have values for the core radius, $a=5\text{kpc}$, and circular velocity of the Sun, $v_{\text{circ}}=220\text{kms}^{-1}$. I start with the dark matter density profile

$\rho(r)={\rho_0a^2}/({r^2+a^2)}$,

from which I can obtain a relation for the mass interior to the radius r

$M(r)=4\pi\rho_0a^2(r-a\text{tan}^{-1}(r/a)+const)$,

$\rho_0=M(r)/(4\pi a^2(r-a\text{tan}^{-1}(r/a)+const))$


To get the local dark matter density at the given radius I assume I need to substitute the above relation into the first equation.

However, I'm also told that the dark halo contributes half of the total mass inside the Sun's orbit but I don't know how to calculate the mass interior to the Sun (probably missing something very simple here). Any help is greatly appreciated.

1 Answers


Brad 05/14/2018.

The density profile given in the question is for a spherical halo, meaning that we can assume the distribution of mass in the halo to be spherically symmetric - allowing us to use the spherically symmetric gravitational acceleration

$g=\frac{GM(r)}{r^2}$,

and the centripetal acceleration for a circular orbit

$a_{centripetal}=\frac{v_{circ}^2}{r}$.

Equating these and rearranging we obtain

$M(r)=\frac{v_{circ}^2r}{G}$,

which can be simply substituted in to the equation for $\rho_0$, which can then be substituted into the density profile given. Using the values given above I get:

$\rho(R_0)=1.85493\times10^{-21}\text{kgm}^{-3}\\\ =0.02726M_{sun}\text{pc}^{-3}$


HighResolutionMusic.com - Download Hi-Res Songs

1 BLACKPINK

Kiss And Make Up flac

BLACKPINK. 2018. Writer: Soke;Kny Factory;Billboard;Chelcee Grimes;Teddy Park;Marc Vincent;Dua Lipa.
2 Martin Garrix

Waiting For Tomorrow flac

Martin Garrix. 2018. Writer: Pierce Fulton;Mike Shinoda;Martijn Garritsen;Brad Delson.
3 Martin Garrix

Access flac

Martin Garrix. 2018. Writer: Martin Garrix.
4 John Legend

Written In The Stars flac

John Legend. 2018. Writer: Kiana Brown;Santoy;Kevin White;Mike Woods;MZMC;The Heavy Group;Rice N' Peas.
5 Martin Garrix

Yottabyte flac

Martin Garrix. 2018. Writer: Martin Garrix.
6 Alan Walker

Diamond Heart flac

Alan Walker. 2018. Writer: Alan Walker;Sophia Somajo;Mood Melodies;James Njie;Thomas Troelsen;Kristoffer Haugan;Edvard Normann;Anders Froen;Gunnar Greve;Yann Bargain;Victor Verpillat;Fredrik Borch Olsen.
7 Cardi B

Taki Taki flac

Cardi B. 2018. Writer: Bava;Juan Vasquez;Vicente Saavedra;Jordan Thorpe;DJ Snake;Ozuna;Cardi B;Selena Gomez.
8 Bradley Cooper

Shallow flac

Bradley Cooper. 2018. Writer: Andrew Wyatt;Anthony Rossomando;Mark Ronson;Lady Gaga.
9 Post Malone

Sunflower flac

Post Malone. 2018. Writer: Louis Bell;Billy Walsh;Carter Lang;Swae Lee;Post Malone.
10 Dyro

Latency flac

Dyro. 2018. Writer: Martin Garrix;Dyro.
11 Zara Larsson

Ruin My Life flac

Zara Larsson. 2018. Writer: Delacey;Michael Pollack;Stefan Johnson;Jordan Johnson;Sermstyle;Jackson Foote.
12 ZAYN

Fingers flac

ZAYN. 2018. Writer: Zayn Malik;Alex Oriet;David Phelan.
13 Halsey

Without Me flac

Halsey. 2018. Writer: Halsey;Delacey;Louis Bell;Amy Allen;Justin Timberlake;Timbaland;Scott Storch.
14 Lady Gaga

I'll Never Love Again flac

Lady Gaga. 2018. Writer: Benjamin Rice;Lady Gaga.
15 Sia

I'm Still Here flac

Sia. 2018. Writer: Sia.
16 Dewain Whitmore

Burn Out flac

Dewain Whitmore. 2018. Writer: Dewain Whitmore;Ilsey Juber;Emilio Behr;Martijn Garritsen.
17 Little Mix

Woman Like Me flac

Little Mix. 2018. Writer: Nicki Minaj;Steve Mac;Ed Sheeran;Jess Glynne.
18 Mako

Rise flac

Mako. 2018. Writer: Riot Music Team;Mako;Justin Tranter.
19 Blinders

Breach (Walk Alone) flac

Blinders. 2018. Writer: Dewain Whitmore;Ilsey Juber;Blinders;Martin Garrix.
20 Rita Ora

Let You Love Me flac

Rita Ora. 2018. Writer: Rita Ora.

Related questions

Hot questions

Language

Popular Tags